Question

which of the following is not an equation that can model the graph below?
none of the answers
all of the answers
y - 1 = 3(x - 3)
y + 2 = 3(x - 2)

Explanation:

Step1: Find the slope of the line

From the graph, we can identify two points. Let's take the points \((3, 1)\) and \((2, - 2)\) (or other visible points). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using \((3,1)\) and \((2,-2)\), \(m=\frac{1-(-2)}{3 - 2}=\frac{3}{1} = 3\).

Step2: Check the point - slope form

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \(m = 3\).

• For the equation \(y - 1=3(x - 3)\): If we use the point \((3,1)\) and slope \(m = 3\), this equation is in point - slope form. Substitute \(x = 3\), \(y=1\) into the equation: \(1-1=3(3 - 3)\), \(0 = 0\), which is true. Also, we can check another point. Let's take \(x = 2\), then \(y-1=3(2 - 3)=3\times(- 1)=-3\), so \(y=-2\), which matches the point \((2,-2)\) on the line.

• For the equation \(y + 2=3(x - 2)\): Rewrite it as \(y-(-2)=3(x - 2)\). Using the point \((2,-2)\) and slope \(m = 3\), substitute \(x = 2\), \(y=-2\) into the equation: \(-2 + 2=3(2 - 2)\), \(0=0\), which is true. And when \(x = 3\), \(y+2=3(3 - 2)=3\), so \(y = 1\), which matches the point \((3,1)\) on the line.

Since both \(y - 1=3(x - 3)\) and \(y + 2=3(x - 2)\) are valid equations for the line (because they satisfy the slope and the points on the line), and the options "None of the answers" and "All of the answers" are not correct. Wait, but the question is which is not an equation that can model the graph. But from our analysis, both \(y - 1=3(x - 3)\) and \(y + 2=3(x - 2)\) are valid. But since the options are given, and if we assume that there is a mistake in the problem or if we re - check, actually, both equations are valid. But since the options are "None of the answers", "All of the answers", \(y - 1=3(x - 3)\), \(y + 2=3(x - 2)\). Since both \(y - 1=3(x - 3)\) and \(y + 2=3(x - 2)\) are valid, the answer should be "None of the answers" because all the given equations (except maybe a mis - option) are valid. Wait, no, let's re - check the slope again. Let's take two other points from the graph. Let's say the line passes through \((3,1)\) and \((0,-8)\) (by looking at the y - intercept). The slope between \((3,1)\) and \((0,-8)\) is \(\frac{1-(-8)}{3-0}=\frac{9}{3}=3\), so the slope is 3. The equation of the line in slope - intercept form is \(y=3x - 8\). Let's check \(y - 1=3(x - 3)\): \(y=3x-9 + 1=3x - 8\), which matches. \(y + 2=3(x - 2)\): \(y=3x-6-2=3x - 8\), which also matches. So both equations are valid. So the equation that is not a model does not exist among the given options, so the answer is "None of the answers".

Answer:

None of the answers